Laguerre's Differential Equation

I need just a little help with this question. I almost solved it except for a minor detail at the end.

Here is the question.

The equation

,

where m is a nonnegative interger, is called Laguerre's differential equation. Show that for each m, this equation has a polynomial solution of degree m. These polynomials are denoted by and are called Laguerre polynomials. The first few Laguerre polynomials are

Ok so I did a power series solution for the problem and arrived at

so

and this is where I get confused.

I see that there is a factorial pattern so I'm trying to use factorial notation to condense.

so would it be

or

So would the final solution be

whatever series is correct?

This solution would show depending what m is chosen that the answer would cancel out certain terms.

Use the results to obtain the 1st few terms in a series expansion about x=0 for a genreal solution for x > 0 to Laguerre's differential equation for n = 0 and 1. Does this just mean plug in 0 and 1 into the general solution I found earlier?